RECURSION Haskell

Hello recursion! Recursion is actually a way of defining functions in which the function is applied inside its own definition. Definitions in mathematics are often given recursively. For instance, the fibonacci sequence is defined recursively. First, we define the first two fibonacci numbers non-recursively. We say that F(0) = 0 and F(1) = 1, meaning that the 0th and 1st fibonacci numbers are 0 and 1, respectively.

Hello recursion! Then we say that for any other natural number, that fibonacci number is the sum of the previous two fibonacci numbers. So F(n) = F(n-1) + F(n-2). That way, F(3) is F(2) + F(1), which is (F(1) + F(0)) + F(1). Having an element or two in a recursion definition defined non-recursively (like F(0) and F(1) here) is also called the edge condition and is important if you want your recursive function to terminate.

Maximum awesome The maximum function takes a list of things that can be ordered (e.g. instances of the Ord typeclass) and returns the biggest of them. You'd probably set up a variable to hold the maximum value so far and then you'd loop through the elements of a list and if an element is bigger than then the current maximum value, you'd replace it with that element. The maximum value that remains at the end is the result.

Let's take an example list of numbers and check out how this would work on them: [2,5,1]. If we call maximum' on that, the first two patterns won't match. The third one will and the list is split into 2 and [5,1]. The where clause wants to know the maximum of [5,1], so we follow that route. It matches the third pattern again and [5,1] is split into 5 and [1]. Again, the where clause wants to know the maximum of [1]. Because that's the edge condition, it returns 1. Finally! So going up one step, comparing 5 to the maximum of [1] (which is 1), we obviously get back 5. So now we know that the maximum of [5,1] is 5. We go up one step again where we had 2 and [5,1]. Comparing 2 with the maximum of [5,1], which is 5, we choose 5.

A few more recursive functions Now that we know how to generally think recursively, let's implement a few functions using recursion.

We will implement take. It takes a certain number of elements from a list. For instance, take 3 [5,4,3,2,1] will return [5,4,3]. If we try to take 0 or less elements from a list, we get an empty list. Also if we try to take anything from an empty list, we get an empty list. Notice that those are two edge conditions right there. So let's write that out:

Quick, sort. We have a list of items that can be sorted Quick, sort! We have a list of items that can be sorted. Their type is an instance of the Ord typeclass. And now, we want to sort them! There's a very cool algoritm for sorting called quicksort. It's a very clever way of sorting items. While it takes upwards of 10 lines to implement quicksort in imperative languages, the implementation is much shorter and elegant in Haskell. Quicksort has become a sort of poster child for Haskell. Therefore, let's implement it here

That's what I'm talking about That's what I'm talking about! So if we have, say [5,1,9,4,6,7,3] and we want to sort it, this algorithm will first take the head, which is 5 and then put it in the middle of two lists that are smaller and bigger than it. So at one point, you'll have [1,4,3] ++ [5] ++ [9,6,7]. We know that once the list is sorted completely, the number 5 will stay in the fourth place since there are 3 numbers lower than it and 3 numbers higher than it. Now, if we sort [1,4,3] and [9,6,7], we have a sorted list! We sort the two lists using the same function. Eventually, we'll break it up so much that we reach empty lists and an empty list is already sorted in a way, by virtue of being empty.

Thinking recursively We did quite a bit of recursion so far and as you've probably noticed, there's a pattern here. Usually you define an edge case and then you define a function that does something between some element and the function applied to the rest. It doesn't matter if it's a list, a tree or any other data structure.